Multivariable Calculus class 110921
Upcoming assignments:
Many students in the class can benefit from the qa's for Chapter 9. I recommend that if you have had any difficulty with Chapter 9 assignments, you at least review them. The disk should also be helpful for many students (see the information at the end of this document for a guide to what's on the disk).
Based on the introduction given in the first two weeks of class, you should be able to make a good attempt on the q_a_ for 10.2. There is no q_a_ for 10.1 (I believe if you click on that q_a_ the link takes you to the q_a_ for 9.7).
It would be to your advantage to have worked through the q_a_ for 10.2 prior to Monday's class.
Sections 10.1 - 10.4 will be assigned for Monday, 10/3/11. The only specific assignments for Wednesday, 9/28/11 will be to prepare for the test on Chapter 9, and complete the q_a_ on Section 10.2 (as an alternative you may, if you prefer, just go ahead and complete the Queries for 10.1 and 10.2, which will be due on 10/3).
In class we talked once more about scalar and vector projection, then sketched a representation of a quadric surface by finding the conic-section intersection of the surface with various planes.
For example, in class we looked at the intersection of a surface with the planes z = 36, z = 0, z = -36, x = 0 and y = 0. We noted the effect of the changing z coordinate on the size of the rectangle which defines the hyperbola in the corresponding plane. Having plugged in a few values of z, we were motivated to see a pattern which will reveal how the figure changes with increasing and decreasing z values.
A similar strategy would work by considering the graphs for a variety of fixed x values (resulting in intersections with planes parallel to the y z plane) and a variety of fixed y values (resulting in intersections with planes parallel to the x z plane). If we can reconcile the various pictures, we should have a good understanding of the 3-dimensional surface.
Note that you must understand the graphing of conic sections, as presented in material you reviewed during the first week, in order to understand what's going on with quadric surfaces.
Guide to the disk:
Unfortunately is wasn't possible to get this information onto the latest disk. You will get another copy with additional files and an HTML program you can use to access them more conveniently. For the present, you can simply browse the disk and access these files.
Given points P and Q in the plane, we find the vector PQ, its magnitude, a unit vector in the direction of PQ and a vector of magnitude 10 in the direction of PQ.
We use a rectangular prism to see why the Pythagorean Theorem extends as it does from 2 dimensions to three.
We apply the Pythagorean Theorem to the points (2, 5, -4) and (x, y, z), and end up with the equation of a sphere centered at (2, 5, -4).
We apply the Pythagorean Theorem to obtain the equation of a sphere centered at (x0, y0, z0) having radius r.
We apply the preceding to obtain the equation of a sphere through (2, 5, -4) with radius 6, and expand the equation into standard form.
We complete the square to obtain the coordinates of the center, and the radius, of a sphere in standard form.
We find the points of our sphere which lie directly above and below the point (2, 6, 2).
We find and sketch the intersection of our sphere with the x-y plane.
Given three points P, Q and R, we ask whether the vectors PQ, PR and QR lie in the same plane, whether the triangle PQR is a right triangle, and what are the angles between pairs of the three vectors.
We consider the vector PQ X PR.
We find the components of a vector F in the direction, and perpendicular to the direction, of another vector `ds. This illustrates the process of scalar and vector projection and two important uses of the projection.
We apply the cross product to find the area of a parallelogram.
We now find the volume of a three-dimensional figure whose base is the parallelogram in the previous clip, sides parallel to vector u:
We develop the equation of a line through the point (x0, y0, z0) and perpendicular to N = a i + b j + c k.